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For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers. Also for questions about quaternion algebras.

The ring of quaternions is a four dimensional division algebra over the real numbers. They are usually denoted as $$\Bbb H$$ in honor of the discoverer, William Rowan Hamilton.

The construction of the quaternions was given by Hamilton as follows: take three symbols $$\mathrm{i},\mathrm{j},\mathrm{k}$$ as imaginary units and define $$\mathrm{i}^2=\mathrm{j}^2=\mathrm{k}^2=\mathrm{i}\mathrm{j}\mathrm{k}=-1$$. As a result, $$\mathrm{i}\mathrm{j}=\mathrm{k}$$, and $$\mathrm{j}\mathrm{k}=\mathrm{i}$$ and $$\mathrm{k}\mathrm{i}=\mathrm{j}$$. Furthermore, $$\mathrm{j}\mathrm{i}=-\mathrm{k}$$ and $$\mathrm{k}\mathrm{j}=-\mathrm{i}$$ and $$\mathrm{i}\mathrm{k}=-\mathrm{j}$$, so $$\mathrm{k}\mathrm{j}\mathrm{i}=1$$.

Another construction of the quaternions was given by Carl Friedrich Gauß as follows: take three symbols $$\mathrm{i},\mathrm{j},\mathrm{k}$$ as imaginary units and define $$\mathrm{i}\circ\mathrm{i}=\mathrm{j}\circ\mathrm{j}=\mathrm{k}\circ\mathrm{k}=\mathrm{k} \circ \mathrm{j} \circ \mathrm{i}=-1$$. As a result, $$\mathrm{i}\circ\mathrm{j}=-\mathrm{k}$$, and $$\mathrm{j}\circ\mathrm{k}=-\mathrm{i}$$ and $$\mathrm{k}\circ\mathrm{i}=-\mathrm{j}$$. Furthermore, $$\mathrm{j}\circ\mathrm{i}=\mathrm{k}$$ and $$\mathrm{k}\circ\mathrm{j}=\mathrm{i}$$ and $$\mathrm{i}\circ\mathrm{k}=\mathrm{j}$$, so $$\mathrm{i}\circ\mathrm{j}\circ\mathrm{k}=1$$.

A quaternion is a linear combination and can represented as versor

$$q=q_{0} + q_{1} \mathrm{i} + q_{2} \mathrm{j} + q_{3} \mathrm{k} ~ \widehat{=} ~ \left[\begin{matrix} q_{0} \\ q_{1}\\ q_{2}\\ q_{3} \end{matrix}\right]\in \mathbb{R}^{4}$$ where $$q_{0}, q_{1},q_{2},q_{3}\in \Bbb R$$

Multiplication between quaternions is carried out by using the distributive rule and the rules for $$\mathrm{i}$$, $$\mathrm{j}$$ and $$\mathrm{k}$$.

The quaternions turn out to be a noncommutative division ring. In fact, $$\Bbb R$$ and $$\Bbb C$$ and $$\Bbb H$$ are the only associative finite dimensional division rings over $$\Bbb R$$. They are also the only normed division algebras over $$\Bbb R$$.